# Calibrated Multiple-Output Quantile Regression with Representation Learning

arXiv: https://arxiv.org/abs/2110.00816 · HTML: https://mlsanity.com/paper/2110.00816/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 55 references that are in the map.
- First posted on arXiv 2 Oct 2021.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Predictive inference with the jackknife+](https://mlsanity.com/paper/1905.02928/) (2019)
- [Distributional conformal prediction](https://mlsanity.com/paper/1909.07889/) (2019)
- [Nested conformal prediction and quantile out-of-bag ensemble methods](https://mlsanity.com/paper/1910.10562/) (2019)
- [Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification](https://mlsanity.com/paper/2011.09588/) (2020)
- [A comparison of some conformal quantile regression methods](https://mlsanity.com/paper/1909.05433/) (2019)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [Flexible distribution-free conditional predictive bands using density estimators](https://mlsanity.com/paper/1910.05575/) (2019)
- [Learning Features of Music from Scratch](https://mlsanity.com/paper/1611.09827/) (2016)
- [Adaptive, Distribution-Free Prediction Intervals for Deep Networks](https://mlsanity.com/paper/1905.10634/) (2019)
- [Improving Conditional Coverage via Orthogonal Quantile Regression](https://mlsanity.com/paper/2106.00394/) (2021)
- [Conformal prediction with localization](https://mlsanity.com/paper/1908.08558/) (2019)
- [Representation Learning Beyond Linear Prediction Functions](https://mlsanity.com/paper/2105.14989/) (2021)

## Built on by (most cited first, 30 of 50)

- [Conformal prediction for multi-dimensional time series by ellipsoidal sets](https://mlsanity.com/paper/2403.03850/) (2024)
- [Minimum Volume Conformal Sets for Multivariate Regression](https://mlsanity.com/paper/2503.19068/) (2025)
- [Conformal Contextual Robust Optimization](https://mlsanity.com/paper/2310.10003/) (2023)
- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)
- [Multivariate Conformal Prediction using Optimal Transport](https://mlsanity.com/paper/2502.03609/) (2025)
- [Optimal Transport-based Conformal Prediction](https://mlsanity.com/paper/2501.18991/) (2025)
- [Conformal Prediction for Class-wise Coverage via Augmented Label Rank Calibration](https://mlsanity.com/paper/2406.06818/) (2024)
- [Rectifying Conformity Scores for Better Conditional Coverage](https://mlsanity.com/paper/2502.16336/) (2025)
- [Lightweight, Uncertainty-Aware Conformalized Visual Odometry](https://mlsanity.com/paper/2303.02207/) (2023)
- [Semiparametric conformal prediction](https://mlsanity.com/paper/2411.02114/) (2024)
- [Deep Non-Crossing Quantiles through the Partial Derivative](https://mlsanity.com/paper/2201.12848/) (2022)
- [Fast Nonlinear Vector Quantile Regression](https://mlsanity.com/paper/2205.14977/) (2022)
- [Conformal Thresholded Intervals for Efficient Regression](https://mlsanity.com/paper/2407.14495/) (2024)
- [Relational Conformal Prediction for Correlated Time Series](https://mlsanity.com/paper/2502.09443/) (2025)
- [Conformal Prediction for Ensembles: Improving Efficiency via Score-Based Aggregation](https://mlsanity.com/paper/2405.16246/) (2024)
- [Multivariate Conformal Selection](https://mlsanity.com/paper/2505.00917/) (2025)
- [Conditional Coverage Diagnostics for Conformal Prediction](https://mlsanity.com/paper/2512.11779/) (2025)
- [Max-Rank: Efficient Multiple Testing for Conformal Prediction](https://mlsanity.com/paper/2311.10900/) (2023)
- [Distribution-Free Conformal Joint Prediction Regions for Neural Marked Temporal Point Processes](https://mlsanity.com/paper/2401.04612/) (2024)
- [Volume-Sorted Prediction Set: Efficient Conformal Prediction for Multi-Target Regression](https://mlsanity.com/paper/2503.02205/) (2025)
- [Multivariate Standardized Residuals for Conformal Prediction](https://mlsanity.com/paper/2507.20941/) (2025)
- [Conformal Prediction for Hierarchical Data](https://mlsanity.com/paper/2411.13479/) (2024)
- [ResCP: Reservoir Conformal Prediction for Time Series Forecasting](https://mlsanity.com/paper/2510.05060/) (2025)
- [Multi-Modal Conformal Prediction Regions with Simple Structures by Optimizing Convex Shape Templates](https://mlsanity.com/paper/2312.07434/) (2023)
- [Deep Ensemble Shape Calibration: Multi-Field Post-hoc Calibration in Online Advertising](https://mlsanity.com/paper/2401.09507/) (2024)
- [JAPAN: Joint Adaptive Prediction Areas with Normalising-Flows](https://mlsanity.com/paper/2505.23196/) (2025)
- [Reliable uncertainty quantification for 2D/3D anatomical landmark localization using multi-output conformal prediction](https://mlsanity.com/paper/2503.14106/) (2025)
- [Multivariate Latent Recalibration for Conditional Normalizing Flows](https://mlsanity.com/paper/2505.16636/) (2025)
- [Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distributions to Multivariate Settings](https://mlsanity.com/paper/2511.15146/) (2025)
- [Learning minimal volume uncertainty ellipsoids](https://mlsanity.com/paper/2405.02441/) (2024)

## Abstract (arXiv)

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller compared to existing techniques.